In the present paper, we will adopt the Dirac-Heisenberg-Wigner (DHW)
formalism of Ref. [1]} and apply it to electrostatic phenomena in
plasmas and vacuum. Specifically, we will reduce the general DHW-system to 4
coupled equations, in the limit of 1D spatial variations. The simplified
system is used to derive a dispersion relation for Langmuir waves,
demonstrating that wave-particle interaction with the quantum vacuum is
possible, leading to electron-positron pair-creation. Moreover, the reduced
electrostatic equations are used to study the influence of perpendicular
momentum (perpendicular referring to the direction of the electric field) on
the process of pair production in vacuum. While the common omission of
perpendicular momentum can be justified to some degree, we point out some
significant corrections introduced by incorporating the full momentum
dependence. Finally, we present our main conclusions and provide and outlook
for future work.
